This thesis summarizes the results of my research during the years of my Ph.
D. at Imperial College London. I was mostly concerned with questions involving
the motivic zeta function for degenerating Calabi-Yau varieties and the monodromy
properties for these degenerations; in parallel to this research, I had a small side
project on simplicial contractions.
The first question we investigated is related to the monodromy property for K3
surfaces which admits triple-points-free models. These Calabi-Yau varieties were
already studied in the literature, the motivic zeta function was explicitly described
and the monodromy property was proven in several cases. In Chapter 3 we prove
that the monodromy property holds for all the surfaces of this family.
The case of triple-points-free degenerations provides the first instance of a family
of Calabi-Yau varieties whose motivic zeta functions can have multiple poles. We
thus decided to construct more examples of such behavior. These examples are
described in Chapter 4 and Section 5.3. Especially interesting is the example in
Chapter 5 since it led us to consider a new problem: the birational invariance of the
motivic zeta function.
The side project on simplicial contraction is described in Chapter 6, we present
a different description of Dupont's simplicial contraction and prove that it coincides
with the classical one, the main advantage of this new definition
is that it may help overcome some of the computational di culties related to
the use of this simplicial contraction.